Widely regarded as one of the greatest mathematicians of all time, Leonhard Euler (1707-1783) made a significant impact on the development of modern mathematics; he also made several important discoveries in the fields of geography, physics, and calculus.
Jeremy Gray is Professor of Mathematics and Statistics at UK’s Open University. He has published several books including The Architecture of Modern Mathematics, edited with José Ferreirós (Seville), on the history and philosophy of modern mathematics, Oxford University Press (2006).
Simply Charly: Much of your work deals with the history of mathematics as a science. What would you say were Leonhard Euler’s most significant contributions to the field?
Jeremy Gray: It is really Euler who showed that the calculus can radically extend our understanding of nature. This was implicit in the work of Isaac Newton, and the idea was being taken up by others before Euler got going, but he opened up many of the applications and created new branches of the calculus to help the process along. We owe to Euler the study of mechanics according to the force of gravity, thought of as an attraction between every minute particle in the system. The study of solid bodies is derived from the study of point masses. Euler gave us the equations of motion of rigid bodies that describe how they rotate—everything from skaters to satellites. He gave us the equations of motion of an incompressible fluid (that is assumed to have zero viscosity). And he re-wrote the calculus of Gottfried Leibniz and Newton, replacing their idea of one variable depending on another with the idea of functions of one or more variables. This is a much more flexible and powerful idea, even if Euler would be surprised at how much it has been generalized since he came up with it.
SC: As an Euler scholar, what first got you interested in his work?
JG: Its importance, first of all. Euler was remarkably prolific, and he dominated every field he touched, so it’s almost impossible to work on the history of science or the history of mathematics in the 18th century without having to look at him. His range is enormous, and then he’s also such a charming writer. He’s keen to show you how he came to his ideas because he wants you to understand them, step by step. You can have the feeling you’re sitting there with him almost having a tutorial as he writes down example after example and steadily builds up his theories.
SC: Euler was one of the most prolific mathematicians of all time. Between losing his eyesight and raising 13 children, how did he find the time to produce over 900 papers? Was this output unusual for a mathematician of his time?
JG: It was extraordinary then, as it is now. Nobody else’s collected works get close. Of course, he does repeat himself—you can get the same material in an article and later in a book. And he’ll write up short observations that other people might have just told a few friends about, but that’s because Euler is following Leibniz and promoting the idea that a mathematician or a scientist should publish what they have found and not keep it to themselves or circulate in the form of a letter. You can argue that it was the force of Euler’s example that helped create the modern world of scientific publication.
His official jobs were in two scientific Academies, the one in St. Petersburg in Russia and another one in Berlin in Prussia, and then back in St. Petersburg. These places were set up because it was believed that the new sciences could be used to modernize the state and make it more productive, so they weren’t like a modern university research institute, but they weren’t like a research division of a major company either. So Euler had periods in his life when he served on commissions and advised on projects, and time when he could get on with whatever interested him. But still, his productivity is unsurpassed.
SC: In addition to his work with mathematics, Euler was also a keen astronomer. What discoveries did he make in this area?
JG: The big problem in Euler’s day was to be sure that Newton’s laws described the motion of the planets, the Moon, and the comets correctly. Although describing the motion of two bodies is easy, describing the motion of three bodies is very hard. In fact, in some sense, it is genuinely unpredictable. So the precise motion of the Moon in its orbit around the Earth is affected by the Sun, the motion of comets is affected by Jupiter, and so on. Euler, and more importantly a young mathematician Alexis Clairaut, were able to show that Newton’s laws do describe these motions well enough. There is no need to alter the inverse square law that Newton had proposed, or to suppose that some other force than gravity is also at work. Euler’s grasp of the mathematics enabled him to give the seal of approval to the work of others, and this opened the way for a French mathematician Pierre-Simon Laplace to do the truly definitive work in explaining the motions of the planets and the Moon in remarkable detail.
SC: Of Euler’s published works, which in your opinion are the most significant?
JG: There are so many! But people often say that the truly great work someone does is to open up a whole new field of study. At that level, I would say the theory of mechanics we talked about at the start—solid bodies and fluids—and the branches of the calculus that went with it: the theories of differential equations, ordinary and partial, based on the concept of a function, are truly significant. Then, in pure mathematics, there’s his re-invention of the theory of numbers. This had been the great love of Pierre le Fermat in the mid-17th century, but few people took it up then. Euler encouraged by his friend Christian Goldbach (of the Goldbach conjecture)—wrote so much on so many different aspects of it that his writings fill four volumes of his Collected Works, and opened the way to the subsequent work of Joseph-Louis Lagrange, Adrien-Marie Legendre, and most importantly Carl Friedrich Gauss.
SC: One of Euler’s claims to fame during his lifetime was his solving of the “Seven Bridges of Konigsberg.” What was Euler’s answer to the problem, and how was it received?
JG: This is a problem about whether you can travel around Königsberg and cross each of its seven bridges in succession without ever crossing a bridge twice. It depends on the arrangement of the bridges—see the picture—and his answer was “No.” He observed that apart from where you start and finish, you must also visit each part of town on two bridges (one on which you arrive on and another on which you leave). So at most, two parts of town can have an odd number of bridges—but Königsberg has four.
Actually, he didn’t think this was a very interesting or important problem, and in his day he was right. But in our time it’s become a hugely important and rather new branch of mathematics, and Euler’s grasp of the key feature of the problem now looks very impressive.
SC: During his time in St. Petersburg, Euler worked on a series of different projects for the Russian government. Did these jobs have any impact on his academic career?
JG: Scholars aren’t sure. He was conscientious, and while in Russia he did his bit to improve the general level of education – he wasn’t at all an elitist about what he did. But he left Russia when life got rather risky for courtiers and academicians and went to Prussia and the Court of Frederick the Great in Berlin. There he did some genuinely applied work on the fountains at the palace of Sans Souci, which was an exercise in hydraulics, and not a very successful one. But Frederick found Euler too sober for his tastes, and they didn’t have much to do with each other. When he went back to Russia he wrote his great, and rather elementary Algebra in German, and I suppose that was rather expected of him. Oddly enough, the book somehow came out in a Russian translation before the German original. But I have the impression that he was mostly treated as an adornment and a person capable of doing useful work at the highest level simply because he was clearly the best person around. And no one was worried about his productivity.
From his 20s, he won international prizes on topics mostly of an applied mathematical kind—he won 13 in all, and adjudicated several more. That can’t have hurt his career.
SC: Were there any classic scholars or mathematicians who had an influence on Euler’s work? Did his work, in turn, influence any modern thinkers?
JG: Euler first studied mathematics seriously under Johann Bernoulli, who had learned the calculus by corresponding with Leibniz. Euler also got on well with several of Bernoulli’s smart relatives, his son Daniel in particular, who did important work on many of the topics that interested Euler. He read Newton’s Principia, of course, and picked up number theory from Fermat’s writings.
As for modern thinkers, he influenced all his contemporaries, and all of the next generation as well, including Gauss. By the middle of the 19th century, his influence was, therefore, being mediated by a number of other writers. But the crucially important function in the study of prime numbers – the so-called zeta function – was introduced by Euler, and its deep importance was first appreciated by Riemann in the 1850s. The Riemann hypothesis about the zeta function is considered by many to be one of the most important problems in mathematics, and is one of the Clay Millennium mathematics prizes with a million dollars offered for its solution. For that matter, the motion of incompressible fluids with viscosity is described by equations named after two 19th century physicists (Claude-Louis Navier and Georges Stokes) and proves that these equations have smooth solutions: it is another Clay Millennium prize, so you can say Euler’s influence is palpable there too. And Euler’s name is rightly attached to quite a few mathematical objects of current interest.
SC: Euler was essentially the creator of the modern mathematical function, making him the bane of high school algebra students everywhere. As a professor of mathematics, how would you explain concepts such as Euler’s identity to a layman?
JG: Well, I’d tease those students a little. There are more frightening things in the intellectual world than the calculus of functions. But basically, Euler’s idea comes in two steps: the input/output idea of a function, and how a function changes as the input changes. The first idea is implemented on every computer you see: give this bit a number and it will give you back a number. How that works is part of the program in the computer, and of course, we needn’t understand the input and output (electric currents as they are) even as numbers. This idea of a function is really fundamental.
Yes, it gets harder when you look at how the output of the function changes as the input changes, especially if, calculus-style, you want to look at the rate of change. But that’s more than compensated for by everything we can do once we master the calculus—the range of things we can understand, the range of things we can make, the places we can go: into space, down to the nanoscale, into mathematical worlds of any number of dimensions.
Euler’s identity is the famous equation \(e^{i\pi} = -1\), which packs an awful lot of the most important symbols in mathematics into a very short space. Let’s look at them one by one, starting on the right. Minus one, negative one if you prefer, is not a number you’ll ever meet as the number of a collection of objects. You might own three cars, or a hundred books, but you’ll never own negative one book. You can tell me about your three cars, but who’s the author of that negative one book? We make negative numbers up so we can deal with debts, so that you can say you have 10 dollars, but you owe Jane 6 dollars, so really you only have 4 dollars to spend. They make perfectly good sense.
The number \(i\) Euler regarded in much the same way. It’s a number we make up so that we can say there are numbers whose squares are negative. You’ll never meet a length like that, but you can use such numbers to solve polynomial equations like \(x^2 = 4, or x^2 -4x + 5 = 0\). Euler called them imaginary numbers because we can imagine them in our minds, and use them according to some simple rules, and (like negative numbers) they’ll never lead you into a contradiction.
OK, \(\pi\) is the half of the ratio of the circumference of a circle to the radius. That leaves \(e\), the basis of the natural logarithms, and for that I’m going to bring in the exponential function that sends \(x\) to \(e^x\). If we apply the calculus, we see that this function grows at a rate exactly equal to its size, which is why we speak of exponential growth.
What Euler saw was that the exponential function can be written as an infinite series:
\(e^x = 1 + x + \frac {x^2}{2!} + \frac {x^3}{3!} + \)….
For any value of \(x\) the terms get very small very quickly, so it makes sense to add them all up. Now, when we say that the exponential function grows at a rate exactly equal to its size we mean that
\(\frac {d}{dx}f(x) = f(x)\),
and you can see that is true of our infinite series.
Now for some trigonometry. If you look at a point going round a circle, its coordinates are \((sin x; cos x)\), and the tangent to the circle at that point is at right angles to the radius so it points in the direction \((cos x; sin x)\), and so
\(\frac {d}{dx}sin(x) = cos(x), and \frac {d}{dx}cos(x) = -sin(x)\).
What are the power series for the functions sin and cos? Well, from the information just
given, Euler saw that
\(sin x = x-\frac {x^3}{3!}+\frac {x^5}{5!}-\)….
and
\(cos x = 1-\frac {x^2}{2!}+\frac {x^4}{4!}-\)….
Indeed, he saw that
\(sin x = \frac {e^{ix}-e^{-ix}}{2i}\) and \(cos x = \frac {e^{ix}+e^{-ix}}{2}\).
So
\(e^{ix} = cos(x) + i sin(x)\),
which is his marvelous formula.
The funny thing is, it seems that he actually never wrote the special case when \(x = \pi\) that we all remember and used to say he did write: \(e^{i\pi} = -1\).
SC: For those interested in studying Euler’s life and work, which sources do you recommend?
JG: That’s easy. Everyone should consult the Euler Archive on the web. It has copies of probably all of his published writings, some in translations, and it will guide you to essays about them. In particular, it will direct you to the splendid columns Ed Sandifer has been writing entitled “How Euler did it,” and they explain a lot. The whole Euler Archive enterprise is very friendly and encouraging.
Then there are the 78 volumes of Euler’s Collected Works (his Opera Omnia). They have excellent, and often very long and informative introductions, in French, German, and English. The Dictionary of Scientific Biography has a very good essay on him by Igor Youschkevitch, and 2007 saw the publication of several books about Euler and aspects of his work because it was the 300th anniversary of his birth. And you can quickly find several more in-depth studies of several aspects of his work.
We don’t have a really good biography of him. The best we have is the one by Emil Fellmann, and Ron Callinger is writing a full-scale biography that he estimates will be published late in 2013. Much more could be done to bring the greatest mathematician and scientist of the 18th century back into public view. He’s very readable, although you might want to take a little time to get used to his style, which isn’t what we’re used to today, and a good way to get started would be to consult Ed Sandifer’s The early mathematics of Leonhard Euler (Mathematical Association of America, 2007). So maybe, as people used to say of him in his day, the answer to your question is: “Read Euler!”.
